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20. Core Formation and Evolution
Core Formation (Terrestrial Planets)
Core formation is the biggest differentiation event in the life of any
terrestrial planet. At least in the cases of Earth and Mars, we know that it is
an ancient event from isotope systematics. In large bodies, it is a high energy
event because of the large energy released from accretion and the energy
released from core formation itself. The process can occur at multiple scales:
migration along grain boundaries, migration of iron along cracks,
segregation of iron in a magma ocean, and diapir descent in the deep mantle.
The above cartoon (from p239, Origin of the Earth and Moon) illustrates the
likely range of processes.
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The next cartoon (immediately above) is from the same book, and intended
specifically for Earth. As explained below it is only part of the core
formation process but it is the one that has received most attention. Here is a
sketch of this scenario for a large planet such as Earth: As the planet forms,
enormous amounts of energy are released, sufficient to melt outer portions
of the planet. The energy available from accretion, expressed as a
temperature rise is
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⎛ M⎞
GM
⎟
ΔT =
≈ 40000K.⎜
RC p
⎝ M⊕ ⎠
2 /3
(20.1)
using terrestrial values for specific heat. Of course, some of this energy may
escape as radiation from the planet surface, but extensive melting seems
unavoidable because much of the accreting mass arrives in big chunks rather
than as a fine “rain” of small particles. Extensive melting and comminution
leads to an “emulsion “ of iron and silicates but the iron can then “rain out”
to a level where the material is less extensively melted. The presence of such
a level is likely because of the steep increase of melting point with depth
within the planet. Descent of iron the rest of the way is then by diapiric flow,
which is geologically fast (one suspects) because the silicates are soft at this
time and the blobs are large. The level at which the blobs form is the depth
of last equilibration for the iron, so the core-forming fluid (and the mantle
left behind)may carry a memory of this pressure and temperature. There is
some evidence to support this picture from a consideration of siderophile
elemental abundances in the mantle. (see next figure, also same book, p283).
These correspond plausibly with the base of a magma ocean ~500 km deep.
The significance of this figure lies in the possible identification of the
pressure (27Gpa) and temperature (~2250K) of last equilibration.
This picture omits the major part of core formation in which giant impacts
deliver large amount of iron at once. These impact simulations (e.g as in the
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223
work of Canup, previous page) suggest that the iron core of the projectile is
broken up at least to the scale of hundreds of kilometers. It is less obvious
whether the iron is broken up all the way to cm scale droplets before
reaching the core. In that sense, part of the core formation process may
involve core merging (the amalgamation of previously formed cores).
Recent data on Hf -W and other isotopic systems suggest a lot of
emulsification following a giant impact but the process may nonetheless
involve imperfect equilibration, as would happen in core merging. (See
Halliday, A.N. Mixing, volatile loss and compositional change during impact-driven
accretion of the Earth, Nature 427 (6974): 505-509 FEB 5 2004). In core merging,
there are giant impacts in which the impacting bodies at least partially
preserve their pre-existing cores. These cores then combine without
complete re-equilibration with adjacent mantle. In this scenario the apparent
“time” of earth core formation can predate the actual time of Earth formation
and the “memory” of the core (in terms of composition) may be that of
events that occurred in much smaller bodies than Earth itself.
(from Stevenson, Nature, v451, p261(2008)
Heating During Core Formation
In addition to the heating that arises from accretion, there is a lot of heat
released as the core forms. Consider the gravitational energy of a
differentiated body relative to that of a undifferentiated body:
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Egrav,undiff = −
3GM 2
5R
M
Gmdm
3GMo 2
5
=−
{1 − x 5 + x 3 [A2 x 2 + (1− x 2 )(A − 2)]}
r( m)
5R
4
0
Egrav,diff = − ∫
Mo ≡
4
πρ R 3
3 o
for our “usual” model of core and mantle (core with radius xR and density
Aρ0 , mantle with density ρ0.). Typically, the change in gravitational energy
is about 0.1 times the total, which is still equivalent to a 4000K or so heating
of everything for an Earth mass.
The consequence of this together with the accretional heating suggests that
the core is likely to form completely molten despite the fact that pressure
freezing could in principle occur.
Core Convection
A terrestrial planetary core, like the mantle, will only convect if it has an
unstable density distribution. If there are no compositional gradients, then
this means that the mean temperature gradient must reach the adiabat. As we
have discussed, the convective state will be close to the adiabat because it
has such a low viscosity. Unlike the mantle, this is not an easy constraint to
meet because it implies a substantial heat flow by conduction alone. The
reason is that the core is a metal and hence a much better thermal conductor
than the mantle.
For a Gruneisen gamma of about unity, the adiabatic temperature gradient in
Earth’s core is about -0.5K/km. If the thermal conductivity is about 3 x 106
cgs, so the conductive heat flow along the adiabat is about
Fcond = -k(dT/dr)ad ≈ 15 erg/cm2-sec
(20.2)
The actual heat flow out of the core, Fcore , might plausibly be given by the
time rate of change of the thermal heat content divided by the surface area.
This cooling of the core is determined by the overlying mantle (mantle
convection).
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Fcore ≈ −[MCpdT/dt]//4πR
2
(20.3)
where M is the core mass and Cv is the specific heat. For a drop of 100K per
billion years, this gives Fcore ≈ 20 erg/cm2.sec, which is uncomfortably
close to the conductive heat transport. Irrespective of the uncertain numbers,
notice that a core can only convect if it is cooling! The most recent estimates
of core conductivity are larger by about a factor of two (see, for example,
Pozzo et al, Nature, 485, p355(2012). This suggests that there is a problem
with getting enough energy to keep the core convecting from cooling alone.
The inner core can help , as explained below, though only for part of Earth
history.
Seismology tells us that the innermost part of the core is solid. While this is
only a small fraction of core mass, it is generally believed that the inner core
is important and perhaps essential to driving outer core motions. The
argument is as follows: Since the adiabat is less steep than the melting curve
of iron (or the pressure dependence of the liquidus of any plausible iron
alloy), it follows that the inner core is a simple consequence of pressure
freezing. But since the core is cooling over geologic time, it follows that the
inner core must grow over geologic time. We can quantify this crudely as
follows. The inner core-outer core boundary is the intercept of the outer core
adiabat with the freezing curve for the core alloy. The freezing curve is
plausibly
Tm(P) = 6000-1700(Pc-P)
(20.4)
where Pc is the central pressure and P is the actual pressure, both in
megabars. The adiabat is plausibly
T = 6000 -1250(Pc -P) -200t
(20.5)
where t is the time elapsed (in billions of years) since the onset of inner core
nucleation. This assumes 200K cooling per billion years. But of course,
hydrostatic equilibrium tells us that
Pc - Pic = 2πGρ2R2/3 ~ 0.25 (Ric/1000km)2
(20.6)
where “ic” refers to the inner core surface. Solving (and taking into account
uncertainty of all the parameters), we get
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Ric ≈ (800 to 1500km).t
1/2
(20.7)
This requires that about 0.8 to 2.5 billion years have elapsed since the onset
of inner core growth because Ric is now 1220km or thereabouts. This is
probably an overestimate since current estimates of core cooling (based on
higher therma conductivity values) suggest faster coling rates and shorter
times for the presence of the core. It is also possible that the difference in
freezing curve and adiabat is smaller than used here. Importantly, inner core
freezing implies two other things of energetic significance; a latent heat
release that currently corresponds to an extra heat flux
2
Flatent heat ≈ 5 erg/cm .sec
(20.8)
and (more importantly) the likely exclusion of some light material from the
inner core, causing compositional convection.
The buoyancy of light, rising material can be very important, as we can see
from the following argument. The inner core is now about 5% of the core
mass. If this is pure iron, then a mass of order (0.1)(0.05) ~0.005 total core
masses of light material has been displaced upwards (recall that the core is
10% less dense than pure iron). In other words, the outer core density has
been reduced by perhaps 0.5% due to compositional effects. But meanwhile,
the core has cooled about 250K. Suppose that conduction alone would have
cooled the core by 500K. Then the compositional convection needs to “heat”
the core (carry hot stuff downward) to the extent of a density anomaly of
250K times the coefficient of thermal expansion ( α = 5x10-6 ). This is 0.1%.
Clearly the compositional convection is able to do it. Of course, this is a
very crude calculation.
Cores cannot convect unless they are cooling. Even when they are cooling,
they may not convect if they do not possess an inner core.
In principle, one can use these arguments for other planets. In practice, the
parameters are sufficiently uncertain that precise conclusions are elusive.
We do not know whether the cores of Mars and Venus (for example) are
cooling. One thing is clear: Core convection is more difficult if the core
cools slowly, and core cooling is slow when there is stagnant lid convection
in the mantle, yet fast when plate tectonics is present. This may be relevant
to understanding the history of core convection and magnetic fields of Venus
and Mars.
In recent years, doubt has been expressed as to whether there is enough
energy from core cooling and inner core growth to drive Earth’s dynamo. As
a consequence, 40K in the core has been suggested (actually an old idea but
now revived). See, for example, The influence of potassium on core and geodynamo
evolution, Nimmo F, Price GD, Brodholt J, et al. GEOPHYSICAL JOURNAL
INTERNATIONAL Volume: 156 Issue: 2 Pages: 363-376 Published: FEB 2004
Core Formation in Icy Satellites
If an icy satellite should start out as a uniform mixture of rock and ice then
energy can be released should the rock settle to the center. This might even
be a runaway process (Friedson and Stevenson, ICARUS 56, 1- 14 (1983).
Viscosity of Rock-Ice Mixtures and Applications to the Evolution of. Icy
Satellites. See question below.
If an icy satellite should accrete as an inhomogeneous mixture (planetsmials
od varying ice/rock ratio) then there is a tendency ot create a bottom –heaviy
ice rock mixture by Rayleigt-Taylor instabilities. This is not able ot convect
becasdeu thermal expansion of cie is small compared to the density
differences arisng from ice-rock mixture variaons. As a consequence, a
doubly-difusive convective regime may arise (O’Rourke and Stevenson,
Icarus, 227,67-77, 2014).
Problem 20.1 Consider a satellite that is half rock and half ice by mass.
Assume a rock density of 3 g/cc and a mean ice density of 1g/cc. What is the
smallest such body for which the temperature rise from core formation is
greater than 200K (plausibly enough to reach the melting point for
reasonably formation temperatures). Assume a mean specific heat of ice of
0.5 cal/g of ice. The rock is 0.15 cal/g of rock. How much larger must it be
to melt most of the ice (latent heat of melting is 80 cal/g of ice)?
Comments: The calorie was defined for water but water ice has only half the
specific heat of liquid water. In reality there must be enough energy from
some other source (accretion or radiogenic) to at least start melting since this
is needed to do the differentiation. Then differentiation will allow the
melting to proceed to completion, as explained in Friedson and Stevenson,
referenced above.